Greater tendoneutralisma
| Interval information |
(Shannon, [math]\sqrt{nd}[/math])
The greater tendoneutralisma is a small comma of the 2.3.13 subgroup which is the amount by which a stack of eight 16/13's minus two octaves falls short of 4/3; that is, it is equal to (16/3)/(16/13)8 and so equivalently also to (13/3)/(16/13)7.
Temperaments
Although the comma is similar in size to something like 81/80, the corresponding temperament is quite accurate because the error can be split evenly over eight 16/13's, so that the pure-3's tuning (very close to 53edo) has 13 off by only 2.78 ¢. A more accurate (lower damage) way of achieving the same (finding 3 by stacking 13's) is by tempering the lesser tendoneutralisma. Very importantly, both are distinct ways of mapping 2.3.13, so that you cannot combine them unless you want to use the trivial tuning of 10edo, so that edos > 10 which have a good 3 and 13 will usually pick between one of these two mappings. A much simpler but relatively much higher error way of mapping 3 for those that prefer sharp fifths is by tempering (16/13)2/(3/2) = 512/507.
Greater Tendoneutralic
Tempering out the greater tendoneutralisma in 2.3.13 leads to the highly notable 10 & 53 temperament, where 10edo is the trivial tuning approximately equal to the pure-13's tuning and 53edo is the tuning practically equal to the pure-3's tuning, although 43edo is an interesting choice for combining this temperament with meantone and 63edo is an interesting choice if you prefer slightly sharp fifths. This temperament is related to submajor, which extends it to the full 13-limit.
Subgroup: 2.3.13
Comma list: 815730721/805306368
Mapping: [⟨1 4 4], ⟨0 -8 -1]]
Optimal tuning (CTE): ~16/13 = 362.248 ¢
Optimal ET sequence: 10, 33, 43, 53, 202, 255f, 308f, 361f
Badness (Dirichlet): 3.037